Likelihood ratio tests are most powerful for testing simple hypotheses and, under monotone likelihood ratio, for testing composite hypotheses; see [1]} and Karlin-Rubin theorem in [2]}. The Karlin-Rubin theorem works with the one-parameter exponential family and Section REF  shows that most powerful testing continues to hold for sequential tests. Even though distributions of sample-size-dependent sufficient statistics ([3]}) belong to a curved exponential family ([4]}, [5]}, [6]}), the distributions conditional on the sample size still belong to one-parameter exponential family. This fact is used in Section REF  to show that likelihood ratio sequential tests are most powerful tests with any pre-determined \(\alpha \) -spending function. This result is applicable to many common group sequential designs.
